There are many ways to disable scrollbars and you'd need to provide more examples than 1 website to cover all cases.
What Bluesky does is not related to overflow CSS. It actually just disables the scrollbars directly with styling, e.g. for Chrome it uses:
body ::-webkit-scrollbar, html ::-webkit-scrollbar {
display: none;
}
-webkit-scrollbar is a pseudo-element that won't work on all browsers, including Firefox from your tags. Firefox actually uses:
body, html {
scrollbar-width: none;
}
to remove it and that's even less compatible. Other browsers might provide different methods.
To reverse it in Firefox on the website itself, you can use:
document.documentElement.style.scrollbarWidth = "initial";
in a userscript or in console.
Crafting a universal anti scrollbar removal script might prove difficult and cause some issues due to edge cases. E.g. some quiz site could hide its answers by hiding scrollbar and you wouldn't want to enable it there then. There could also be some cases of some more JS-focused scroll blocking that might be harder to cover, but not sure they're your target.
So, to sum up, it's up to you to combine info from this and more overflow focused answer(s) and craft something that suits your needs accurately.